If you haven't played with this new animation already, then please do so. Click the upper left button until it disappears. Notice while you're doing that that at some point the growth of the population of pink love spheres seems to slow down. When the upper left button goes away, click the lower right and view the graph.

In our original model back on page 1 the rate of growth of the population was directly proportional to the size of the population. This new model is also fairly simple, but it takes into account that there are limits - that rates of growth will be affected by the size of the population trying to access finite resources. Therefore we should somehow take into account that each individual's chances to reproduce are going to be hindered by the size of the population with which each individual is competing. Previously the rate of growth, DP / Dt, was set equal to kP, where k was a constant interpreted as the average number of spawn resulting from each individual per generation. Now we'd like this number k to decrease as the population P gets bigger. Ths simplest way to do this is to replace k by (k - bP). That is, set

DP / Dt = (k - bP)P = kP - kbP2.

What I've done in the animation above is to set k = 0.5, as before, and set b = 1/1600. So

DP / Dt = (800 - P)P/1600.

Therefore as P grows from 1 towards 800, the rate of growth DP / Dt first increases, peaks at P = 400, then gets smaller and smaller, and by the time P gets to 800 the rate of growth has shrunk to zero (click here for a discussion of derivatives and extreme values). Zero growth means a stable population, so 800 is the highest population attainable. This is indicated by the horizontal green line that appears in the graphing part of the animation. The exponential growth curve is also shown. I haven't graphed the actual theoretical curve for the solution to this new differential equation, because it would be too big a pain in the neck, and I don't want to do it. I don't have to, and I'm not going to.

Philosophical Note: Highest attainable populations are not nice places to be. It's very crowded, and socially narrow and brutal. That's bad.